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Math Help - Determinants of this matrix - pls help

  1. #1
    Member zzizi's Avatar
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    Determinants of this matrix - pls help

    Hi, can someone help me out with this, how do I solve this matrix question:

    Let A = [a,b,c; d,e,f; g,h,i] Given that detA= alpha and alpha is not equal to zero, express each of the following terms of alpha:

    a) det(2A)

    b) det(A^-1)

    c) det [2a,2b,2c; d,e,f; g,h,i]

    d) det [a,b,c; d,e,f; g,h,i]

    e) det [a,b,c; d-3a, e-3b, f-3c; g,h,i]


    Thank you
    Last edited by zzizi; May 15th 2012 at 02:54 AM. Reason: typo
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  2. #2
    Member Goku's Avatar
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    Re: Determinants of this matrix - pls help

    Did you mean det(A)= alpha
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  3. #3
    Member zzizi's Avatar
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    Re: Determinants of this matrix - pls help

    Yes, sorry that was a typo - I have corrected it. Thanks
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  4. #4
    Member Goku's Avatar
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    Re: Determinants of this matrix - pls help

    det(A) = a(ei - fh) - b(di-fg) + c(dh-eg) = alpha

    Now:

    det(2A) = 8a(ei - fh) - 8b(di-fg) + 8c(dh-eg) = 8(a(ei - fh) - b(di-fg) + c(dh-eg))= 8alpha

    try the rest...

    check Determinant - Wikipedia, the free encyclopedia
    Last edited by Goku; May 15th 2012 at 04:03 AM.
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  5. #5
    Member zzizi's Avatar
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    Re: Determinants of this matrix - pls help

    Thank you Goku,

    I have these answers, are these correct?

    a) 8alpha
    b) 1/alpha
    c) 2alpha
    d) alpha
    e) alpha
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  6. #6
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    Re: Determinants of this matrix - pls help

    Hello, zzizi!

    These are easy if you know the properties of determinants.


    Let A = \begin{bmatrix}a&b&c \\ d&e&f \\ g&h&i \end{bmatrix}

    Given that |A| = \alpha\text{ and }\alpha \ne 0, express each of the following terms of \alpha:

    (a)\;|2A|

    Answer: . 8\alpha




    (b)\;|A^{-1}|

    Answer: . \frac{1}{\alpha}




    (c)\;\begin{vmatrix}2a&2b&2c \\ d&e&f \\ g&h&i\end{vmatrix}\right|

    Answer: . 2\alpha




    (d)\;\begin{vmatrix}a&b&c \\ d&e&f \\ g&h&i\end{vmatrix}

    Answer: . \alpha




    (e)\;\begin{vmatrix}a&b&c \\ d-3a & e-3b & f-3c \\ g&h&i\end{vmatrix}

    Answer: . \alpha

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  7. #7
    Member zzizi's Avatar
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    Re: Determinants of this matrix - pls help

    Thank you very much for your clear answer. How did you type it up like that?
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  8. #8
    Member zzizi's Avatar
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    Re: Determinants of this matrix - pls help

    Im a little stuck with this one, how do I work out this determinant? can anyone help with this?

    Thanks

    \begin{vmatrix}a &b  &c \\d-a&e-b  &f-c \\g-2a&h-2b  &i-2c\end{vmatrix}
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  9. #9
    Member Goku's Avatar
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    Re: Determinants of this matrix - pls help

    Use the property det(AB) = det(A).det(B)...

    AB = \begin{vmatrix}a &b  &c \\d-a&e-b  &f-c \\g-2a&h-2b  &i-2c\end{vmatrix}

    Let B = \begin{vmatrix}a &b  &c \\d&e  &f \\g&h  &i\end{vmatrix}

    we already know det(B) = alpha

    Now find a matrix A multiplied by B,that will give you AB...

    I will leave this to you to do, if you have trouble finding sucha matrix then post again.
    Then Find det(A) multiply by det(B)...

    I have not done this in a long time, so please excuse if there are any mistakes.
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  10. #10
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    Re: Determinants of this matrix - pls help

    Hello, zzizi!

    I'm a little stuck with this one. .How do I work out this determinant?

    . . \begin{vmatrix}a &b  &c \\d-a&e-b  &f-c \\g-2a&h-2b  &i-2c\end{vmatrix}

    How do you work out any determinant? . . . Just crank it out!

    However, we can use some Properties on this one . . .


    Here's an important property of determinants.

    If a multiple of one row is added to (subtracted from) another row,
    . . the value of the determinant is unchanged.


    Suppose we have: . \begin{vmatrix}a&b&c \\ d&e&f \\ g&h&1\end{vmatrix}. . Its value is:. (aei + bfg + cdh) - (afh + bdi + ceg) .[1]

    Subtract row-2 minus row-1: . \begin{vmatrix}a&b&c \\ d-a & e-b&f-c \\ g&h&i \end{vmatrix}

    Subtract row-3 minus twice row-1: . \begin{vmatrix}a&b&c \\ d-a & e-b&f-c \\ g-2a & h-2b & i-2c \end{vmatrix}

    This determinant has the same value . . . the value at [1].

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  11. #11
    Member zzizi's Avatar
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    Re: Determinants of this matrix - pls help

    Is the number one that you put in the bottom right hand corner of the determinant delibrate?


    Suppose we have: . . Its value is:. (aei + bfg + cdh) - (afh + bdi + ceg) .[1]

    And is the determinant alpha once again?
    Last edited by zzizi; May 16th 2012 at 05:34 PM.
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  12. #12
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    Re: Determinants of this matrix - pls help

    another way to see that subtracting 3 times row 1 from row 2 does not change the determinant:

    let P be the matrix:

    P = \begin{bmatrix}1&0&0\\-3&1&0\\0&0&1 \end{bmatrix}

    then

    PA = \begin{bmatrix}a&b&c\\d-3a&e-3b&f-3c\\g&h&i \end{bmatrix}

    (verfiy this!)

    since det(PA) = det(P)det(A), to find det(PA), we need to calculate det(P).

    det(P) = (1)(1)(1) + (0)(0)(0) + (0)(-3)(0) - (0)(1)(0) - (0)(0)(1) - (1)(0)(-3) = 1 + 0 + 0 - 0 - 0 - 0 = 1

    therefore, det(PA) = det(A) = α

    (this was what Goku was hinting at his post).

    you can see that the value of det(P) won't change if we replace "-3" by "r", and furthermore, it really doesn't matter "where" r goes (as long as it's not on the main diagonal), because there will always be some other 0 on any diagonal (or "extended diagonal", like when you "loop around") containing r to "cancel it out". so subtracting r times any row from a DIFFERENT row, will never change the determinant of a 3x3 matrix.
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  13. #13
    Member zzizi's Avatar
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    Re: Determinants of this matrix - pls help

    Thank you very much for the explanation
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